In 2018, Li, Liu and Peng [1]}
solved this problem completely.
They
gave the corresponding improvement on
the result of the signless Laplacian spectral radius
in Theorem REF .
Interestingly, the extremal graphs in their theorems are quite different from those in
Theorems REF .
Recalling the definition in (REF ) and (REF ),
we denote \(X=\lbrace v\in V(H_{n,\delta }) : d(v)=\delta \rbrace ,Y= \lbrace v\in V(H_{n,\delta }): d(v)=n-1\rbrace \) 
and \(Z=\lbrace v\in V(H_{n,\delta }): d(v)=n- \delta -1\rbrace \) .
Let \(E_1(H_{n,\delta })\)  be the set of those edges of \(H_{n,\delta }\) 
whose both endpoints are from \(Y\cup Z\) .
We define
\( \mathcal {H}_{n,\delta }^{(1)} =\left\lbrace  H_{n,\delta } \setminus E^{\prime }: E^{\prime } \subseteq E_1(H_{n,\delta })~\text{with}~|E^{\prime }|\le \lfloor {\delta ^2}/{4}\rfloor \right\rbrace . \) 
